The Basic Concepts of Set Theory
1
2
3
4
5
Symbols and Terminology
Venn Diagrams and Subsets
Set Operations and Cartesian Products
Surveys and Cardinal Numbers
Infinite Sets and Their Cardinalities
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Chapter 1
Section 2
Venn Diagrams and Subsets
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Venn Diagrams and Subsets
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•
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Venn Diagrams
Complement of a Set
Subsets of a Set
Proper Subsets
Counting Subsets
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Venn Diagrams
In set theory, the universe of discourse is called the universal set, typically
designated with the letter U.
Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these
diagrams, the universal set is represented by a rectangle and other sets of interest
within the universal set are depicted as circular regions.
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Venn Diagrams
The rectangle represents the universal
set, U, while the portion bounded by the
circle represents set A.
A
U
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Complement of a Set
The colored region inside U and outside the circle is labeled A' (read “A prime”).
This set, called the complement of A, contains all elements that are contained in U
but not in A.
A
A
U
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Complement of a Set
For any set A within the universal set U, the
complement of A, written A', is the set of all
elements of U that are not elements of A.
That is
A  {x | x U and x  A}.
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Subsets of a Set
Set A is a subset of set B if every element of A
is also an element of B. In symbols this is
written A  B.
B
A
U
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Example: Subsets
Fill in the blank with  or 
 to make a true
statement.
a) {a, b, c} ___ { a, c, d}
b) {1, 2, 3, 4} ___ {1, 2, 3, 4}
Solution

a) {a, b, c} ___
 { a, c, d}
 {1, 2, 3, 4}
b) {1, 2, 3, 4} ___
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Set Equality (Alternative Definition)
Suppose that A and B are sets. Then A = B if
A  B and B  A are both true.
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Proper Subset of a Set
Set A is a proper subset of set B if A  B
and A  B.
In symbols, this is written A  B.
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Example: Proper Subsets
Decide whether  ,  , or both could be placed
in each blank to make a true statement.
a) {a, b, c} ___ { a ,b, c, d}
b) {1, 2, 3, 4} ___ {1, 2, 3, 4}
Solution
a) both
b) 
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Counting Subsets
One method of counting subsets involves using a tree diagram. The figure
below shows the use of a tree diagram to find the subsets of {a, b}.
a subset? b subset? 4 subsets
Yes
No
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Yes
No
{a, b}
{a}
{b}
Yes
No

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Number of Subsets
The number of subsets of a set with n
elements is 2n.
The number of proper subsets of a set with n
elements is 2n – 1.
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Example: Number of Subsets
Find the number of subsets and the number
of proper subsets of the set {m, a, t, h, y}.
Solution
Since there are 5 elements, the number of subsets is 25 = 32.
The number of proper subsets is 32 – 1 = 31.
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